## University Physics with Modern Physics (14th Edition)

We require that $\Phi(\phi)=\Phi(\phi+2 \pi)$. $\Phi(\phi+2 \pi)=e^{im_l(\phi+2 \pi)}= e^{im_l\phi} e^{im_l(2 \pi)}= \Phi(\phi) e^{im_l(2 \pi)}$ For the desired expression to be true, it is required that $e^{im_l(2 \pi)}=1$ Euler's formula states: $e^{im_l(2 \pi)}=cos(m_l(2 \pi))+i\;sin(m_l(2 \pi))$ Therefore, an inspection of the cosine graph reveals that $e^{im_l(2 \pi)}=1$ if and only if $m_l$ is an integer.